on the sectional irregularity of congruences

manuscripta math. 90, 239 - 266 (1996) manumcripta mathematica

Sprlnger-Vet'la8 1996

O n t h e s e c t i o n a l i r r e g u l a r i t y o f c o n g r u e n c e s

Luis Giraldo and Ignacio Sols* Departamento de Algebra, Fac. Matem~tticas

Universidad Complutense de Madrid 28040 Madrid, Spain

Received March 21, 1996

0 Introduction Let G = G(2, 4) C ps be the grassmann variety of lines in the projective space pa over an algebraically closed field of characteristic 0. A congruence is an integral surface X C G.

The Chow groups are: Aa( G) = rJ3Z

G) = ,j2Z

A I ( G ) = rllZ

The cycle qa is a special linear complex, i.e. parametrizes all lines in pa meeting a given line; the cycle r h is an c~-plane, i.e. parametrizes all lines in pa passing through a given point; the cycle r/~ is an a'-plane, i.e. parametrizes all lines of pa contained in a given plane; the cycle r h is a line pencil, i.e. parametrizes all lines contained in a given plane and passing through a given point of it. Therefore

!

7]3 2 = r ] 2 + ~ 2

!

q3'r/2 =q3"r/2 = r h

q3 " ql = pt

zl2 ~ =z l ; 2 = p t !

rl2 �9 rl~ = 0

and the Chow class of a complex, i.e. threefold of G, is given by its degree or intersection number with a generic line pencil; the Chow class of a surface of G, is given by its bidegree (d, d'), i.e. the intersection number d with a generic

* Partially supported by CICYT PB90-0637

240 Giraldo, Sols

a-plane, and the intersection number d' with a generic c~'-plane; the Chow class of a ruled surface of p3, or curve of G, is given by the intersection number of the curve with a generic special line complex which is clearly the degree of the corresponding ruled surface in pa. The genus of this curve is the irregularity of the surface. Even when surfaces of G are usually called congruences, in this paper a congruence will denote just an integral surface in G.

We define the sectional irregularity q of the congruence to be the genus of the curve in G that we obtain as intersection Y = X f3 p4 with a generic p4 C ps, i.e. it is the irregularity of the corresponding ruled surface in pa.

The problem of finding good bounds for the hyperplane section genus of a congruence is an interesting one, as it is crucial both in their classification and in tile study of the distribution of the bidegrees. Concerning the first question we can cite the recent works of Herns and Sols ([10], [l 1]), Arrondo and Sols ([1], [2]) and Gross ([4], [6]) and also those of classical geometers as Kummer, Fano and Roth. The distribution of the bidegrees of smooth surfaces in G was also studied by Mark Gross ([5]) using as an essential tool a bound for the sectional irregularity.

Then, the need of deriving sharp bounds for the sectional irregularity, both in terms of the bidegree (d, d') and in terms of the total degree d = d + d' is clear. In this paper we obtain such bounds and prove the following

T h e o r e m Let X be a con9ruenee of bidegree (d,d') and degree d = d + d' not contained in a complex of degree < s

I. I f d > 2s(s + 1) then the ruled surface consisting of all lines of the congruence X meeting a general line of p3 has irregularity

where

with

q <_ B(d, s)

d_.~+s(s_3) d B(d, s) = 1 + 2,

~+s(s-3)~ 1 + 2*

( s - r ) ( ~ - ~ + � 8 8 i f t is odd

4--;- 2 if t is even

d = s t - r , O _ < r < s

2. If d > s(s + 1),d' > s(s + 1) then, the ruled surface consisting of all lines of the congruence X meeting a general line of p3 has irregularity

wh e re

q < B(d, d', s)

d d B{d,d ' , s ) = 1 + -~(s + -s - 4) -

r ( s - - r ) ( s - 1 )

2s

with

d' d' r'(s - r ' ) (s - 1) +1 + ~-(s + --s - 4) - 2s + min{d,d '} - 1

d = st - r, O < r < s and d' = st' - r', O < r' < s

On the sectional irregularity of congruences 241

The bound B(d,s) was conjectured by Arrondo and Sols in [2]. The numerical character of the points obtained after intersecting the congruence with a general cone was used there to get the bound, translating to their s tudy the previous work due to Gruson and Peskine in [9]. But the congruence in [2] is necessarily supposed to be contained in a complex of degree s. Then, the natural and important open problem is that of proving that the conjectured one for congruences not contained in a complex of degree less than s holds. This is a difficult problem whose solution has required a different approach, turning to the study of the set of points that appear after intersecting the congruence with general c~ and c~'-planes and carefully determining the relationship between their numerical characters. This approach has lead to the bound in terms of the bidegree B(d, d',s) and also to the essential point without which the bound in terms of the total degree d cannot be obtained.

Concerning previous bounds in terms of the bidegree, very little was known before Mark Gross gave in [6] for a congruence not contained in a linear complex and whose triadic locus is of dimension _< 2 the bound

q _< ( d - 1)(d' - 1) + ~ ( d - 1 ) ( d - 3d')

His approach to this problem is completely different to ours. If we compare this with our bound B(d,d',s) for s = I, we find that Gross' is bet ter when d' is close to 3d and ours is bet ter when d is close to d'.

The distr ibution of the paper is as follows. After proving in section 1 a Laudal 's type lemma (lemma 1.1) that we need for the sequel, we devote the second section to prove that the bound in B(d,d') holds. Furthermore, there are congruences of bidegree (d, d') not contained in a complex of degree < s for which this is an equality if and only if

�9 either: I d - d ' I<_ 1

�9 or: s = 2 and d,d' are odd with I d - d' I= 2

We describe all such extremal congruences in the third section (prop. 3.1). In part icular all of them are ar i thmetical ly normal of postulation s (i.e., the minimum degree of a complex containing those congruences is s). In the process of description we find (lemma 3.4) the relation between the Gruson-Peskine character o[ the intersections of an ar i thmetical ly normal congruence with an c~-plane and an a ' -plane, which we think is the most interesting feature of this research.

From the description we derive in prop. 3.2 that for all those values of d,s and s for which the bound is sharp, there are smooth congruences reaching the bound.

Finally, we prove in section 4 that the bound in 1 holds and from the relation that can be established between the bounds in terms of the degree and the bidegree, we can also conclude that the former is sharp when d = - 1 , 0 , 1 (rood 2s) and that it is not sharp otherwise. If the congruence is arithmetically normal then a refinement holds and is sharp (even in the smooth case), so we conjecture this is the sharp bound for all congruences.

We wish to thank Mark Gross for his careful reading of this manuscript.

242 Giraldo, Sols

1 A L a u d a l ' s t y p e l e m m a

We derive in this section a Laudal's type lemma (see [12]) proving it in the line of [7].

Let X be a congruence whose order, i.e. the degree of the intersection Z of X with a general a-plane P~ (with x C p3), is d.

L e m m a 1.1 If Z is contained in a plane curve of degree s, with s(s + 1) < d, then X is contained in a complex of degree s.

Proof. Let F C pa x G be the incidence variety with obvious projections.

F

p3

q �9 G

We know that PicF = Z | Z. Let P be q - l (X ) . By hypothesis, the intersection of F with a general fibre P~ of p is included in a general plane curve of degree s which must be unique. Thus

p.q*:[x(s) = p.Zr(O,s) 7 ~ 0

(in fact it is a line bundle as 2-r ~ OF and p.OF ~- Opa). Furthermore, Vx E p3, if Z~ = F n P~ and ~ < s, h~ = 0. Otherwise p.Zr(0,s) would be a torsion sheaf, which is impossible as

p . I r (0 , a) ~ Opa = p.OF

For some t E Z, it is

o r h~ = h~ ~)) we consider t minimal with this property. The hypersurface S of F of bidegree (t, s) containing F so defined is integral as P is.

Since q has fibres of dimension 1 and d i m s = 4, the image of S by q has dimension 3 or 4.

If the dimension of the image is 3 there is a complex of degree s containing X, which finishes the proof. (Observe that t = 0 in this case).

Otherwise the image of S has dimension 4, i.e.

S q.-A~ G

is dominant. The main point is that q does not contract the dimension of S, while the three-dimensional subvariety F C S is mapped by q onto the surface X C G, so its dimension is contracted. Since X cannot be hidden in an irreducible component of S not dominating G because S is integral, it can only happen that I' is contained in the singular locus of the morphism q. This singular locus is

the support of the torsion sheaf f~s/c on S presented by the second fundamental sequence

0 ~ O s ( - s , - t ) --+ ~F/a | Os -~ ~s/c --+ 0

It is easy to check that 9ts/a = OF( l , - -9) . Therefore, the restriction of this

sequence to the fibre p-l(x) of F ~ pa at a general point x E pa, which is the general a-plane P~ C G, is

0 -~ o ~ ( - s ) -~ o~.(1) --+ ( ~ / c ) | K(x) -~ 0

where S~ = 5 A p-'(x) (this is a curve of degree s in p-'(z) = P~ which is irreducible since S is and x is general) and K(x) is the residue field of the point. Since the set of points Z = FFlp-~(z) is contained in the support of ~s/a | which is seen in the last sequence to be the intersection of the curve S~ C P~ of degree s with another plane curve of degree s + 1, we conclude that d = degZ < s(s + 1), a contradiction.

R e m a r k 1.1 The bound of the former Iemma is sharp:

The smooth congruence X,+I where a general section

0 -+ O -+ E(s + 1) --~ "/x,+, (2s + 1) -+ 0

of the twist s+ l of the universal bundle E (see [2]) on G vanishes has order

d = - ( s + ~) + (s + ~)~ = s(s + ~)

On the other hand

H~ : H~ E) : 0

(in fact s+ l is the postulation of Xs+l), while for a general a-plane P~, we have

0 --, tf~ : H~ �9 Opt ( -1 ) ) -+ H~

thus the restriction X,+I N P~ lies in a plane curve of degree s.

2 B o u n d o f t h e s e c t i o n a l i r r e g u l a r i t y i n t e r m s

o f t h e b i d e g r e e

Let X be a congruence of bidcgree (d, d') not contained in a complex of degree < s. Let Z = X N P~ and Z' = X 0 P~, be the set of d,d' distinct points of intersection of X with a generic a-plane and a'-plane. Define a >_ s as the minimum degree of a curve E of p2 containing Z and define (7' >_ s analogously.

Let 7r : E --+ p1 be the projection of E from a point of p2 outside E and let 7r'ZZ,Z : {~),=0c'-! Op1 (--hi), with no >_ na _> �9 �9 _> n~- i _> a. We call this sequence the character of Z and denote it x(Z).

Applying rr. to the sequence

244 Giraldo, Sols

we obtain

0 --* Zz,r. --+ O~ ~ O z --* 0

a--I a--1

0 ~ { ~ O p , ( - - n i ) --* {~ Op,( - - i ) --+ 7r.Oz --* 0 i=0 i = 0

which in terms of graded modules over the graded ring R = K [ X o , X 1 ] of p1 is

a - 1 a - 1

o -~ �9 R(-n,) -~ � 9 1 6 2 --, x ~ o i=O i=O

The base of $,=o~'-~ R ( - i ) is 1 , z 2 , x ~ . . . . , x~-~;and x~ expresses in this base through the equation of the curve E (not passing through P=(O,O,1), i.e. including the monomial x~ in its equation).

If S is a curve of degree s >_ a containing Z and choosing P out of E and of S we can also apply :r. to the sequence

and get

0 "-'* ZZ,S --~ O s -* OZ "--* 0

s - 1 s - 1

0 "--+ 0 0 p l (--T/2j) ~ 0 0 p l ( - - j ) ~ 71".0 Z ~ 0

j = o j = o

of corresponding graded modules

s - 1 s - 1

0 ---. (--[-) R ( - r n j ) --, @) R ( - j ) ~ M 0 j=O j=O

Comparing with the former presentation of this module through the split sequences

0 0

s - 1 s - 1

�9 R(-~) - - �9 R(- i ) i=o" i=o"

n-1 s - 1

o ~ O R ( - m A ~ O R ( - J ) ~ x ~ o j=o j=o

a - 1 a--1

o -- On(-"d-- On(-i)-- x -- o i----0 i=0

0 0

we obtain that

s--1 a - 1 s - 1

~.zz,~ = O o p , ( - m ~ ) = O o P , ( - . , ) e O o p , ( - i ) j = 0 i = 0 a

The schemes X N P ] when we vary the choice of c~-plane p2 form a fiat family over p3 so the proposition of [7] p. 224 applies to conclude that the character is connected, i.e.

[ n i - n i + l I <_ 1

Observe that d = P ~ - i) and Z--, i = 0 k

o'--1

ht(Zz(l)) = ~ ((rz, - I - 1)+ - (i - 1 - 1)+) i = 0

where for a C Z, we have that a+ = a if a >_ 0 and a+ = 0 otherwise. ! > ' > n t a , _ l ( ~ f i t ) . Analogously, we obtain a connected sequence n o _ n l . . . _

Let L be a generic line of p3 and let C3 be the 3-dimensional nondegenerate cone of G with vertex L C G whose points correspond to all lines meeting L. Observe that C3 is the intersection of G with the hyperplane p4 tangent to G at L and that it contains a px family of a-planes and another of a'-planes. Let Y = C3 fq X. We want to show that the curve Y has arithmetic genus h i (Or) bounded by the expresion in the introduction.

A generic a-plane P~ of C3 is a generic a-plane of G and P I n Y = P ] r ' I x = Z. Analogously, P~, fq Y = pz, n X = Z'. Observe that P~ U P~, is a hyperplane section of C3 C p4, so we have

0 ~ O c t ( - 1 ) ~ Oc~ ~ Op~upi , ~ 0

thus

0 ~ Zw,c3(l - 1) ---+ Zw, c3(/) ---* Zzuz,,p~upi,(l) ---+ 0

Therefore,

q h'(Oy) h~(zy, c3)<_~ ' = = h (Zzoz, p~Up~o,(t)) / > 0

with equality if and only if hl(Z~.,c~(t)) = 0 for all t C Z. We say in this case that Y is arithmetically normal.

Let ~(l) be the restriction map in

o -11~ r2oUP2o,(l))~ H~ e H~ r~o,(l)) ,(t! HO(OL(t) )

/

Hl(Zzuz,,p2 up2 ,(1)) ~ Hl(2"x,p2o(l)) @ HI(Zz,,p2o,(I))

246

this is

Giraldo, Sols

q < ~ h l ( Z z , p ~ ( l ) ) + ~-~hl(Zz,,p~,(l)) + ~-~(h~ - d i m i m ~ ( l ) ) 1<0 I>0 />0

We bound now these three terms. From

0 ~ O p 2 ( - - a ) --+ I Z , p ~ ~ I Z , E "-~ 0

we see that a = m i n { l / h ~ r 0}, thus a >_ s from the l emma we proved in section 2.

Fur thermore , we get for all l > 0

h ' ( Z a v ~ ( l ) ) = h'(Za,~(l)) - h~(Op~(l - o))

= hl(Tr,Ez,z(1)) - h~ " - l - 3))

Therefore,

o'-I

= ~ h ' ( O e ~ ( l - rid) - h~ - l - 3) ) i = 0

a - - I a- -1

= ~ 2 ( ~ , - l - 1 )+ - E ( i - l - 1 )+ i = 0 i=O

h'(Zz , p~(l)) = V• />0

where for a character or connec ted nonincreas ing sequence ); : no _> nl _> . . . _> n~_l(_> a) of length a we denote

O'--1

vx = Z Z ( ( ~ , - l - 1 )+ - ( i - l - 1 ) + ) l>0 i = 0

We also call the degree of the charac te r d(x) to

a--1

d ( x ) = ~ ( ~ , - i) i=O

L e m m a 2.1 Let X : no >_ nl >_ . . . >_ n o - l ( _ > ~r) be a character of length c and

degree d, and let 6 > d. Let ~ : no >_ . . . >_ no_2 >_ n~- i >_ . . . >_ no_l(_> 5") be a prolongation of X to a character ~ of longer length 6 and degree 5.

' > >_ ' be a character X' of the .same length cr as X and Consider n o . . . . n~,_ 1 degree 5, such that X' > X, i.e. n} >_ ni for all 0 < i < ~ - 1 and one of these a inequalities is a strict one. Then

Proof:

�89 <v~ ,

~--1 rt,

i=a j=i

On the sectional irregularity of congruences

< ( n u m b e r of t e r m s ) x ( h i g h e s t p o s s i b l e v a l u e of ( j - 2)+ for t h e m )

= (~ - d)(n, - 2)+

a n d

a - I nl~

v~, - ~ = E E (J - 2)+ / : 0 n , + l

>__ ( n u m b e r of t e r m s ) x ( m i n i m u m p o s s i b l e va lue of ( j - 2)+ for t h e m )

2 4 7

> (~ - d ) ( n . _ , - ~)

a n d ( n ~ _ , - 1) - (n~ - 2 ) > 1 w h i c h g i v e s V:? < V x. []

L e m m a 2 . 2 Given a character x : no > . . . > n , _ l ( > a) of length a and degree d, and given 5 >_ d there is a character X' >-- X of the same length a and degree

Proof : We c a n a s s u m e , by a t r i v i a l i n d u c t i o n , t h a t ~5 = d + 1. If ni = n for

all i, t h e n we c a n t a k e

I X : n + l > n > _ . . . > _ n

If, o t h e r w i s e , we h a v e t h a t ni = n for c ~ - 1 > i > k a n d n, > n for 0 __ . . . > nk+1 > n _> . . . _> n

t l h e n we can t a k e all n', = n, e x c e p t t h a t n k = nk + 1, i.e.

[]

! X : n o > _ . . . > n k + l > _ n + l > n > _ . . . > _ n

R e m a r k 2 .1 In particular, i f x is connected we can obtain a connected X'.

L e m m a 2 . 3 For every connected character X of length s and degree d, it is

n 'h e re . . . . { ' }

Xs,d : m t ; r r t i

where d = s r - c , 0 _< c < s; d ' = sT' -- ~', 0 <_ e' < s, ] r -- r' I<_ 1, and

m i = ( s + r - 1 ) - i - 1 O < i < s - 1

m , = ( s + 7 - 1 ) - i s < i < s - 1

rn'~= ( s + r ' - 1 ) - i - 1 O < i < e ' - I rn' , = (s + r ' - 1) - i g ' < i < s - 1

2 4 8 Giraldo, Sob

Proof. It is a purely combinatorial result that can be found in [8]. []

Now we can derive upper bounds for the first and second terms in the above bound of the sectional irregularity: Let X = x ( Z ) = {n,}0<~<,_~ (thus with d(x) = d and length ~ _> s). Let X" be its t runcation at leve ls , i.e. X ~ : n o >_ . . . ~ T/s--I.

Clearly" d(x ~) < d. By lemma 2.2, there is a character X" -> X ~ of the same length s as Xs and of degree d. We then have by lemma 2.1 applied to X" and lemma 2.3 applied to ~*'

l>O

The second term

is bounded analogously.

h' (zz,,e 5 (l)) l>O

R e m a r k 2.2 Observe for the sequel that if there is an equality, then V• = Vx,,, thus X = X s', i.e. the congruence is in a complex of degree s, and furthermore )~ ~ m a x

Xs,d �9

We prove now that the third term is bounded by

y ] h ~ - dim ( i rnr <_ d - 1

l>o

so it will be analogously bounded by d' - 1, thus by

min{d,d '} - 1

concluding the proof of the theorem. It is clear tha t irng'(l) contains imp( l ) in the sequence

0 ~ EI~ - 1)) ~ H~ ~(~l) H~

n ' ( z a p 2 o ( l - 1 ) ) - I1'(zap,o(l)) ~ o

Therefore

~ . ( h ~ - dim i m ~ ( l ) ) < ] ~ ( h ~ -- dim i m ~ ( l ) ) = />0 />0

E ( h ' ( 2 " x , p ~ ( l - 1 ) ) - h l ( I z , p g ( l ) ) ) : />0

h~(Zz) = d - i

as wanted.

3 S h a r p n e s s o f t h e b o u n d s for l oca l l y C o h e n - M a c a u l a y c o n g r u e n c e s

We provc in this section that our bounds are sharp if I d - d' I_< 1 or I d - d' I= 2, s = 2 and d, d' are odd, and that these very bounds are not sharp, (i.e. the inequality is strict) in all remaining cases.

D e f i n i t i o n 3.1 An equidimensional surface X of G is called ari thmetically normal (a.n. for short) ff

hi(Z,\-a(l)) = 0 f o r all l E Z and i = 1,2

L e m m a 3.1 Let X, X be locally Cohen-Macaulay surfaces of G that are linked in the sense that their schematic union is the complete intersection of two complexes of G. Then X is a.n. if and only i f ) ( is.

Proof. It is analogous to the proof for linked schemes of codimension 2 in P~, since again Hi(Oa(l) ) = 0, for 0 < i < d imG. A detailed proof can be found in

[2] []

L e m m a 3.2 A congruence X of G is arithmetically normal if and only if its intersection Y = X N C3 with a linear complex p4 N G is arithmetically normal.

Proof. This follows inductively from the exact sequence

f t t ( Z x , a ( l - 1 ) ) -+ IfX(Zx,a(l)) ~ H'(Zy,p ,nc( l ) ) -+ H2(Zx ,a( l -1) ) --+ U2(Zx,c(l))

and from the vanishing of all these cohomology spaces when l > > 0 or l < O. []

L e m m a 3.3 Let X be an equidimensional surface in G of bidegree (d,d') and such that its numerical bicharacter is

x(Z) ; x ( Z ' ) = no > n , >_ . . . > ~ _ ~ ; n ' 0 > n ' l _> . . . _> n ' r

IJX1 is the surface linked to X by the complete intersection of two complexes of degrees M and N, then

�9 The minimal resolution of the ideal sheaf Zz

m--] + 0 --+ 0 (.gp2(--Vi) --+ Op2(--#j ) --+ (.9p2 --+ 0 z --+ 0

i=1 3=1

can be determined from the {ni}. (Analogously b r Zz, and {n'i}).

* Given minimal resolutions of the ideal sheaves Zz and Zz, the numerical bicharacter is determined.

250 Giraldo, Sols

�9 If Z1 is the set of points where X1 meets a general a-plane, the minimal resolution of the ideal sheaf Zz, is determined in terms of the one of Zz. (Analogously for Zq) . Then, its numerical bicharacter is also.

Proof. It is enough to prove it for Z C p2. First of all observe that Z C P~ is a closed subscheme of codimension 2 which is projectively Cohen-Macaulay (see [3]).

Then we have that there is a minimal projective resolution

m--1 ra

0 - . | o p t ( - . , ) --, | o i . ~ ( - ~ , ) -+ oe~ -~ o ~ --+ 0 (1) *=1 j----1

(all homomorphisms are of degree 0) Recall that m is the minimal number of generators of 2"z and /~a are the

degrees of a minimal system of homogeneous generators for "2-z, and it holds tha t gjm=~ ~ j = EF=~ ~ ,,,.

Suppose now that a = postulation o f Z, and let C, be a degree a curve in p 2 with Z C C, . We have an exact sequence

0 --+ O p [ ( - - o ) ---+ ~ Z , P ~ ---+ ~Z,C~. --'+ 0 (~)

Let x (Z) : no _> nl >_ . . . _> no_~(_> c 0 be the character of Z. Define for all t E Z +

H(t ) = # { i / n , = t}

a ~ ( t ) : H ( t ) - H ( t - i ) , / ' ,+( t ) : ( A ( t ) ) + , S ( t ) : A + ( t ) - A ( t )

Then a simple cohomological calculation taking into account both sequence (1) and sequence (2) yields

m = E A+(t)+ I tEZ+

and if #1 _< #2 -< . . . -< #m then they correspond to [Zl : o, and {tt/}j=2 ...... is the set { t /A+(t ) > 0} and each value t appears A+(t) times in that set, In part icular , #2 = n ,_ l .

We can in the same way realize that {u,},=l ...... is the set { t /S ( t ) > 0} and that each value t appears S(t) t imes exactly.

R e m a r k 3.1 The fact that a minimal resolution of a module is unique up to the addition of a resolution of the module 0 corresponds in our situation with the fact that we can define"eztended character" by means of a non-minimal curve.

We can also see that minimal resolution and character determine each other (let #1 <_ I1~ <_ . . . <_ #m be the degrees of minimal generators for Zz then

= postZ = # , , n , _ , = # 2 , a n d # { i / n i = t} = #{ i /u i > t + 1} - # { i / # , > t + l } , t > _ n , _ l )

As can be seen in [13] if Z1 C P~ is the subscheme linked to Z by a complete intersection of degrees M and N, M < N, then Oz~ has a projective resolution

0 --, o m ( - ~ j ) -* G O m ( - ~ d - - , o m --, oz, ~ o j = l i = 1

with 5j = M + N - I t j , % = M + N - u i , i = 1 , . . . , m - 1 and%~ = N, 3'm+1 = M. This resolution is not necessarily minimal, but we can get a minimal one

by jus t disregarding repeated factors in the resolution. Being the initial one a minimal resolution (so gi r ui, Vi , j ) , the only possibly "extra" factors are the ones corresponding to "~m = N, 7m+1 = M, but this will be the case if and only if N or M equal the degree of a generator of Zz (i.e., equal pj for some j )

Anyway, we conclude that the character of Z1 is

�9 M, N 7~ # j ,Vj . Let ~1, _< "}2 <_ . . . _< "},~+1 be a ordering of the {~ti}. Then 1 = 72 and x ( Z , ) : n ~ > n I > > n ' (>_ cq), where a, =71 , n~_,

- - - - �9 . . _ _ f f l - - I

# { i / n } = t} = •{i/6i > t + 1} - #{ i / S i >_ t + 1}

�9 In case M = #j or N = #k (or both) then we obtain a minimal resolution of the form

p p + l

o -~ @ o i ,2 ( -g , ) --, G o r , 2 ( - - ~ , ) --, Op2 ~ oz , j = l i = 1

- - ,0

which {~j} and {•i} are perfectly determined. Then the character is in obtained as before.

L e m m a 3.4 Let X be a locally Cohen-Macaulay, arithmetically normal surface of G of bidegree (d ,d ' ) and postulation (i.e. minimal degree of a complex containing X ) s.

Let Z, Z' be the intersections with an c~-plane P~ and an cd-plane P],. Let

X; X'

" 0 ~ 0 ~ . . . ~ n a _ l ; n t o ; . . . , nla*-i

be the characters of Z and of Z'. Write d = " r a - p,O <__ p < a and d' = r ' a ' - p',O <_ p' < or', and let t = m i n ( r , r ' ) + 1. Assume for instance G' > a. Then

I. s - l < _ a , a ' < s

2. [ n , - n ' i [ < 1 f o r a l l O s( s + 1) then s = cr = a'

4. I d ' - d [ ~ s

252 GirMdo, Sols

5. Assume X is integral and d,d' > s(s +. l ) . Then s = a = a', and for a complex C of degree s containing X it is h~ 7 ~ 0

Proof: Let C be a complex of degree s containing X and let L C Q4 -- C be a point outside C. Let C3 be the three-dimensional cone of Q4 with vertex L (i.e. the intersection in p5 of Q4 and the tangent hyperplane to Q4 at L). Let C3 be the blowing-up of Ca at the vertex and let

~r : 53 = P(OQ2 | OQ2(-1 , - -1) ) --* Q2

be its canonical projection to the smooth quadric surface @.. We denote by Y and S both the intersections X fl C3 and C ~ Ca and their isomorphic transforms in C3. Since no generator of C3 may be contained in S, the restriction

has fibres of length s, thus the first and second line bundles on S in the sequence

0 --* Zy.s ~ Os ~ Or --* 0

have as images by functor 7r. vector bundles of rank s on Q2

s--1

0 -~ 7r.Zy, s ---* 7r.Os = ( ~ O 0 . ~ ( - i , - i ) ~ 7r.Oy ~ 0 i = 0

Since S does not meet the exceptional divisor of the blowing-up, the line bundle Os(1) = OQ,(1) | Os is 7r'OQ~(1, 1) thus, by projection formula

= l)

Recall that Y is arithmetically normal by lemma 3.2. Therefore, for all l E Z

0 : H'(Zy,c3( t ) ) ~ H ' (Z~ ,s (O) --" H~(Oc'3(-s + l)) : 0

and being 7r : S ~ Q2 of finite fibres thus with null functor Rl~r., the Leray spectral sequence degenerates and we obtain

0 = f I ' (Zv , s( l)) = Hl(~r.(Zv,s(l))) = Hl((Tr.Zv, s)(l, l))

By a theorem that can be found in [2], we conclude that

s - I

= @ O Q 2 ( - m ; , - m j ) , with atl fro; I<__ i i = 0

Let P~ and P~, be two meeting lines of Q2 whose counterimages by 7r are an c~-plane P~ ,and an a '-plane P~,. Then Z = P~ n Y consists of d points in the plane curve S~ = P~ fl S of degree s(_> a) and analogously Z' consists of d' points in the curve S=, of degree s _> a ' in P~,. If s = a = # . then (1) is obvious. Otherwise, assuming for instance a ~ >__ a, it is s > c~

0 ~r.Zz,so ~ ~r.Oso ~ ~r.Oz " 0

s-I s-1 OOp,o(-J)-- o

3=0 j=O

and analogously for rr.Zz,so,. Recal l ing what we poin ted out in at the beginning of the preceeding sect ion it is

s-I c~--I s-I @ C0p~(-#nj ) = O O v l (--Y~,l)�9 O O v l (--i) (the ~qecoJ~d facto," is not zero) 3=0 j=O d=r7

s-1 a'--I s--1 (~) O v , o , ( - m ' j ) : ( ~ O p , ( - n ' i ) e ( ~ O p , ( - i ) (the second factor may be zero) 3:0 j=O a'

Since all n, >_ ~r, it is m i n { m j } = G, i.e. a factor O ( - G , - c r - e) wi th e 6 { - 1,0, 1 } occurs in the above decompos i t ion of 7r.Zy,s.

From

fI~ ca(c~ + 1)) --* H~ s(a + 1)) --* H'(Oc~(~r + 1 - s)) = 0

and

H~ § 1)) D H~ - r r 0

we obta in tha t H~ c~(a + 1)) # 0. Since X is a r i thmet ica l ly normal , we conclude tha t s _< ~r + 1 < or' + 1 which is (1).

The second factor in the above decompos i t ion of 7r.(Zz,so ) is zero or O p , ( - s + 1) depend ing on whether G = s or a = s - 1, and the analogous remark holds for v,.(Zz,so,). Therefore, since all I rnj - m ' 3 ]< 1 we conclude tha t ] ni - nti [_< 1 for all 0 < i < cr - 1, and if ~7' = cr + 1, it is n'~,-i = a'. This proves (2).

Passing to the long exact sequence of cohomology from the last sequences we see tha t d = ,-1 d' P s - l ( m " E i = 0 ( r n j - j ) and = z~j=0t , - J ) , s ~ ] r n , - m ' j 1_< 1 implies (4).

Finally, (5) is noth ing but l emma 1.1. []

Examples:

1. Let X be the (2, 2) s m o o t h a r i thme t i ca l ly normal congruence where th ree generic sect ions of the t angen t bundle to the g rassmannian are dependen t . Its charac ter is 2; 2 thus ~ = c~' = 1, and its pos tu la t ion is not 1, as could be guessed naively, bu t is 2. It also can happen tha t -not real izing how subt le the s i tua t ion is- one might expect t ha t being

254 Giraldo, Sols

7r.Zz,So = Op, ( - 2 ) and ~r.Ez,,So, = Op, ( -2 )

(in the notations of the former proof) the decomposition of 7r.Ev, s should contain the factor OQ2(-2 , -2 ) . This is not true, since actually

2.

rr.Zy, s = OQ2(-1 , - 2 ) �9 0 0 2 ( - 2 , - 1 )

As an example that even smooth arithmetically normal congruences can have c~ r a ' is enough to think of an a-plane or in the Veronese congruence (described at the end of this paragraph), or of any spinorial congruence, i.e. defined by a generic section of a positive twist E'(n) of the universal subbundle E' . From the nullity of Hi(E' (n) ) for 0 < i < 4 it can be easily deduced that its character is

2n - 2 , 2 n - 3 , . . . , 2 n - (n - 1 ) = n + l ,n; 2 n - 2 , 2 n - 3 , . . . , n + l , n , n

which exemplifies the assertions of the former lemma.

3. Let X be a complete intersection of complexes of degrees s,t. Then X is arithmetically normal.

4. Let X of bidegree (d, d'), with for instance d' > d, be the rest of a surface ~" of bidegree ( d ' - d , 0) in the complete intersection of a linear complex and a complex of degree d ~ (i.e. this complete intersection is the schematic union of X and d' - d planes of type a). Then X is arithmetically normal if and only if ] d' - d I< 1 (indeed it is equivalent to the arithmetical normality of X as can be seen in examples 1,2).

Linkages in Q4 by the use of two complexes of degrees ll,l~ are linkages in p5 by three hypersurfaces (of degrees 2,11,l~). It is remarked in [13] before proposition 4.1 that its assertion holds also in codimension bigger than 2, thus we obtain the lemma below. A finer form of this lemma can be proved, but the following is all we need for the sequel

L e m m a 3.5 Let X be a locally Cohen-Macaulay, generically complete intersection, arithmetically normal congruence and let 11,12 be integers that are bigger or equal than all the integers of its character. The generic complete intersection F h N Ft~ containing X is the schematic union ("geometrical linkage") of X with another congruence Xx which is smooth (thus irreducible since hl(Zx~) = O) if X is smooth.

Proof. Just observe that for l bigger or equal than the integers of the character of X, t is ha(:Yz(1)) = hl(Zz,( l)) = 0, thus h2(2"~.(l)) = 0 and ha(Y.x(1)) = O, so h~ = X(Zx(l)) , i.e. / is bigger or equal than the degrees of a minimal set of generators of X or, in more geometrical terms, the congruence X is cut by all the complexes of degree l containing it.

Therefore, 11,12 as in the statement work for Prop. 4.1 in [13] concluding the lemma. []

On t h e sec t ional i rregular i ty o f congruences 255

Assume now that X has genus reaching our bound. Then the generic section Y = X 71 C3 is ar i thmetical ly normal , since all inequalities in the proof are equalities, so in part icular

h~(Zy,~) = ~ h'(Zzuz,,p~op~,(t)) />0

Therefore X itself is a r i thmet ica l ly normal , by the lemma. Fur thermore , the congruence must be contained in a complex of degree s by

the remark in section 3, and has character X: = X~,$"

x~ j ~ = { ~ ; ~ ' , }

w h e r e d = s r - e , 0 < e < s ; d ' = s r ' - e ' , 0 < s ' < s , ] r - r ' ] _ < 1, and

m i = ( s + r - 1 ) - i - 1 0 < i < e - 1 m i = ( s + r - t ) - i e d. By l emma 3.5 the rest of X in the generic complete in tersect ion of a l inear complex and a complex of degree d r is a smooth congruence of bidegree (d' - d, 0), i.e. an a -p lane or the empty congruence.

Assume from now on tha t X is conta ined in a complex C of min imal degree s > 1 which must be integral since X is. By the lemma 3.4

h~ r o

Therefore a complex of degree t intersects C in X plus another divisor X1 in C. Assume first tha t r = r ' so tha t t = r + 1 = r ' + 1. Then the character of the

surface XI is

S, r ; 8, r r

The nonconnectedness of the character brings no contradict ion since X1 may be not integral. Therefore I r - r ' ]_< 1 , and assume for instance r ' < r.

Since the locally Cohen-Macaulay surface X1 is contained in the complete intersect ion of a quadra t ic complex F2 and a complex F, of degree s, the rest is a surface X2 of bidegree (s - r + 1, s - r ' + l) lying in a linear complex, and so it. (:an be linked to an a~-plane or tile empty congruence. In this case we have I ~ l - d ' t_< 1

Second case: r r 7'. Assume for ins tance r ' = r + i. Then it is clear - l emma 3.4- tha t r < r' . After linkage by a complex of degree s and a complex of dcgree t and taking into account lemmas 3.1, 3.3 and 3.4 we conclude that r _< 1

a n d r ' = s - 1 .

�9 In case r = 0 we conclude that X is linked to an a ' -plane.

256 Giraldo, Sols

�9 In case r = 1 we can see by means of one further linkage and the con- sideration of the lemmas cited above that the only possible case is s = 2 and the character of X1 is 1;2,2. Therefore it is a nondegenerate locally Cohen-Macaulay surface of bidegree (3, 1) (if it were an integral one, then [11, 2] it would be a Veronese surface) It is clear that in this case we have d ~ - d = 2, and having linked by complexes of degrees 2 and t tile degrees d = 2 t - 3 , d ~ = 2 t - 1 are odd.

Summariz ing we have

P r o p o s i t i o n 3.1 Let d, dt, s be integers such that d, d' > s ( s + l ) . Assume for instance d' > d. Let X be a locally Cohen-Macaulay integral congruence of bidegree d, d' not contained in a complex of degree < s whose sectional irregularity reaches our bound. Then X is arithmetically normal and is contained in a complex of degree s and

1. either d = d', in which case

�9 sld and X is the complete intersection of two complexes of degrees s

a n d

�9 s does not divide d. X is the rest in Fs n FIll+ 1 of the complete

intersection X1 = / : ~ n F([~l+l)s_a.

2. or d' - d = 1, in which case

�9 = arise then:

- s[d' ( i .c . ,r ' = 0). X is the rest of an a- plane in the complete intersection r~ n F([~]+, ) .

- s does not divide d'. X is the in FsNF([~]+I ) of the res tXa in

F l n F([dj]+l)s_ d of an a'-plane.

�9 [~} = [d] + l ands id . In this case X is the rest in F, nF([~]+l ) of

the rest, X i , of an a'-plane in F1 N F~.

3. d' - d = 2, then s = 2, d and d' are odd and X is the rest in 1:2 n F[~]+ 1

of a nondegenerate (3, l) surface.

Next we prove the converse existential proposit ion

P r o p o s i t i o n 3.2 Let d, d ~, s be integers such that

1. d , d ' > s ( s + l )

2. we have either

�9 I d ' - d ] < l or

�9 t d ' - d l = 2 , d andd ' are odd a n d s = 2 .

Then, there is a smooth arithmetically normal congruence contained in a complez of degree a and not contained in a complex of degree < s, and such that its sectional iregularity is

q = B(d, d', s)

Proof: We invert the process in the proof of Proposition 3.1. Call n = [~] for shorter

1. Case d = d'. We can distinguish two possibilities

�9 s]d Then we just consider the general complete intersection X = F', r3 1,'~. Then we know that Ix .G has a presentation (corresponding to a minimal free resolution of (gx) given by

0 - ~ o c ( - s - ~ ) --+ o c ( - s ) | O G ( - - ) --, I x , a - , 0

from which is clear from lemma a.3 that the character of X corre- m a x . e m a x sponds to X,.s , X,,d, �9 Of course, X is a.n.. Finally. we will prove tha t

its irregulari ty is B(d, d, 8). But this is a straightforward computat ion a s

q = h ' ( O x n c a ) = h~(Ixnca,ca) = ha(--T'X(-1)) - ha ( Ix )

s does not divide d: The generic complete intersection X1 = F1 R F(~+I),-~ is smooth, a.n. and of character (n + 1)s - d; (n + 1)s - d. Since both s and n + 1 are bigger or equal than (n+ 1)s - d , the rest of X1 in the general complete intersection /~ O F~+I is smooth (see [2]), thus integral, of bidegree (d, d) and of character . . . . . . . . X,.~ , X,,d according to lemma 3.3. It is also (lemma 3.1) a.n. since a complete intersection is a.n.. Let us call it X. In order to see that its sectional irregularity is B ( d , d , s ) we recall from [2] that if the ideal of a surface S has a presentat ion by locally free sheaves

O-+ s ~ M - + Is--+ O

and ~' is linked to .5" by complexes of degrees 11.12, then Z 9 has a presentat ion

0 ---, .a .4v(- i , - l~) --, Z :V( - l , - t~) e O o ( - l , ) �9 O o ( - l ~ ) --, I~ .o --+ 0

Therefore we have a presentation for Zx

o - , o o ( - s - , ~ ) < ~ o o ( - ~ . - 1 ) --, O o ( - k ) e O c ( - s ) ~ O o ( - , ~ - l ) --, :Zx.G --, 0

258 Giraldo, Sols

w h e r e - k = n ( s - 1 ) - d . We can thus calculate q as we did in the previous case from this sequence recalling that the dualizing sheaf wc is O c ( - 4 ) and that

h~ = 5 - 5 5 = 0 i f m < 5

We obtain B(d , d', s) as wanted.

2. C a s e d ' = d + l .

�9 Suppose sld'. Then n = [~] = [s

\~'% take an a-plane, X1 and consider its rest in the generic complete intersection Fs N Fn, call it X. Then, as X1 has character 1; , the character of X is determined by lemma 3.3. As we saw before, being (see [21)

0 ~ O h ( - 1 ) --* E ~ Zx , .a ~ 0

a presentat ion for the ideal sheaf of the a-plane, we have a presentat ion

0 ~ E ( - s - n + l ) ~ O c ( - s - n + l ) | 1 7 4 ~ I x , c ~ 0

as E v = E(1). Now we observe that X is smooth, a.n. , has maximal character and we can calculate q and show that it is B(d , d', s) from the preceeding sequence and the knowledge of the cohomology of E (see [21) ,f does not div,de We have thon = =

Two possibilities arise:

- W'e first consider that sld. We can proceed as before and we will get a smooth, a.n. congruence of maximal character and of irregulari ty q = B(d , d', s): let us consider an a ' -p lane X2, take its rest in the general complete intersection F1 n Fs and denote it by X1. One more linkage of XI by complexes of degrees s and n provides a rest X which is a congruence as the one we were seeking for.

- If s does not divide d, s tar t with an a ' -plane, X2, take its rest, Xx, in a general complete intersection F1AF(~+l)s-e and again the rest of X1 in the general compete intersection Fs Cl F~+l. Call it X. X is smooth, a.n., of character Xs~,~; X,,'~5, A completely analogous calculation shows that its sectional irregularity is B(d , d', s) .

3. Case d' = d + 2 , d o d d a n d s = 2. Let X1 be the V e r o n e s e s u r f a c e o f P S . It is easy to see [2] that it is contained in a smooth quadric, which we can take to be the grassmannian of lines in pa embedded via the Plficker embedding. It is nondegenerate. It can also be seen in [2] that its ideal sheaf can be presented by the sequence

0 -+ O a ( - 2 ) a --+ E ( - 1 ) | E ( - 1 ) --+ __Txl,c ~ 0

From this is clear that X1 is a.n. and also that its character is 2, 2; 1

Since [~] + 1 >_ 2, the rest X of X1 in the generic complete intersection F2 N F([~]+I ) is smooth, thus integral, a.n. of bidegre (d, d') (both d and

d' are odd) and character X~,~; X,~,~, ~. Finally, its sectional irregularity is q = B(d, d', 2). [ ]

4 B o u n d o f t h e s e c t i o n a l i rregu lar i ty in t e r m s o f t h e d e g r e e

The goal of this section is to establish a bound B(d, s) for the sectional irregularity of congruences of degree d and not contained in a complex of degree < s. We will assume that d > 2s(s + 1).

Let us first consider a congruence X of degree d and postulation s, i.e. contained in a complex S of minimal degree s. As a~ > 2s(s + 1) this complex is unique and integral.

Let Ux be the integral variety parametrizing all 2-dimensional cones of G = Q4 C ps whose vertex is not in S and whose intersection with X consist of distinct points, in distinct generators, and not contained in a surface of pa of degree < s. It is nonempty by lemma 4.1 and lemma 4.2 that we will prove later.

Consider in Ux x Q4 the incidence subscheme C2 and its intersections S and with Ux x S and Ux x X . All of them are schemes over Ux. We assume u C Ux scheme theoretic point (not necessarily closed) given and denote by C2 D ~ D Z the fibres of these schemes over u. Then s is a curve over K = K(u) cut by a surface of pa of minimal degree containing Z. The vertex of the cone C2 is not in Y;. Let

re : 02 = P (Op , G) (.gp, ( -2 ) ) ---+ Q, ~ p1

be the blowing-up of C.2 at the vertex and its natural projection. Clearly, every fibre of re intersects s in a scheme of finite length s. Observe that Os| = (.9~: | 7r'(.gpl(2/). Applying functor r . to the sequence

we get

0 --+ Zz, s -," Or. -+ O z -~, 0

s - 1 s - 1

0 �9 Op, �9 op, (-2i) 02 '-- ,.o2 - , 0 i = 0 i----0

with fi0 > fil >_ . . . >_ fi,-l(:> 2s). The sequence {fi,} and the Hilbert polynomial h~ determine each other since, for all l e Z

260 Giraldo, Sols

s - 1

h'(Z,,o,(l)) : h' : :

i:0

D e f i n i t i o n 4.1 We call fii the numerical character associated to the congruence X and to the point u 6 Ux, If u is a general closed point of Ux wc call it the numerical character of the congruence. This is also the numerical character associated to the generic point of Ux, since it has the same associated Hilbert polynomial than the one associated to the general closed point of Ux.

T h e o r e m 1 For the numerical character {fii} of a congruence, it is for all i <_ s - 2

fi{ - ni+~ _< 2

Proof: Let u G Ux be the generic point, and dualize the above associated sequence

s - 1 s - 1

o --, ( 9 o p , (20 ~ ( 9 op, (n,) ~ M ~ 0 i : 0 i = 0

with .A4 -- gxtXOr.O2, Opt). This exhibits generators a0 , . . . ,as-1 of A/l of degree - r i o , - . . , - n s - 1 . Let

Adi C jt4 be generated by ao , . . . , ai. Since A,4 is torsion free on the integral scheme Z, then each .Mi is supported in the whole of Z.

If for some 0 2, then a general linear equation x 6 H~ not vanishing at the vertex of the cone, when acting as 3/l :5~ Ad restricts to M i ~ .M~ since for all j < i, it is deg xaj = 2 - fij < -f i i+l

Therefore

i

xaj : E ~ kak k = 0

so that

de t (x I - (ajk)) = 0

vanishes on Z. This is a homogeneous equation including the monomial H}=0aj3 which is in H~174 since cuj 6 H~174 = H~174 (2)). It also includes the monomial x i E H~ | Op3(i)). We see clearly that the degree of this homogeneous equation is i < s - 1, so it defines a surface of p3 vanishing in the support 2 of .Mi, which is a contradiction. []

L e m m a 4.1 The intersection of an integral congruence X of degree d with a general cone C2 in Q4 consists of d distinct points, not two of them in the same generator of C2.

On the sectional irregularity of congruences 26"1

Proof: The intersection with the general special-linear complex is a ruled surface Y of p3 whose generators meet a line L of pa. Let L v be the p1 of planes containing L. Let P1, P2,- . . be points of L in which the rule of Y passing through them spans with L the tangent plane to Y along all the points of the rule. Let C be the curve of Y consisting of points lying in more than one generator, and let Q1, Q~,.-. be the points of intersection of L with C. Consider a pro]ectivity L ~, L v such that none of the points P1 ,P2 , - . .Q1,Q=, . . . is assigned a plane containing a generator passing through it. The lines M of p3 meeeting L and contained in the plane ~y(M 93 L) are parametrized by a cone C2 of Q4 with the two desired properties.

The fact that no generator of Y passing through P~ is contained in ~(P{) makes it sure that the d points of X N C ~ are distinct. The fact that no generator of 1/passing through Qj is contained in ~(Qj) assures us that no couple of points of X C) C2 lie in the same generator of C2. []

L e m m a 4.2 Let d ,a be integers such that d > 2a(a + 1), and let X be an integral congruence in Q4 of degree d. If the intersection of X with a general cone C~ C Q4 is contained in a surface of Pa(D C2) of degree a, then X is contained in a complez of degree a.

Proof: This result is proved in [2] []

We come now to derive the announced bound. Let X be a congruence of postulation a(>_ s). If we let ]4' be a (necessarily irreducible) complex of degree a containing X, we choose a point L G G such that L r I42 and consider a cone in G with vertex L: we get an irreducible curve W = W ~ C~ of class (2a, a) in Pic 02 containing the points Z = X n C2. As the morphism C2 ~ p l is of finite fibres when restricted to W, we have

Let us recall that

{=0

q = Z z - - -< E hI(Z2. dl)) ( 3 ) 1>1 / > I

with equality if X is a.n., and

: h1(Z ,w(l)) - h (Oe2(2(l-

so the numbers {7~i}0 ....... 1 completely determine the bound in (3). Let us denote x(Z) the character of the points 2, and V• the number we got in 3.

Def in i t ion 4.2 Let "r be the minimum positive integer such that there exists an irreducible curve [fV C C2 containing ]. cut out by a degree r surface in p3 not passing through the vertex of the cone.

Then r wilt be called the homogeneous postulation of the points 2 .

262 Giraldo, Sols

R e m a r k 4.1 It is clear that r < a and also that I;V gives an irreducible (2r, r ) curve in C~.

Lemma 4.2 shows clearly tha t , being d > 2s(s + 1) we have r _> s (this would be t rue even in case tha t 2 is conta ined in a reducible curve of type (2r, r ) ) .

Projec t ing from the vertex we ob ta in

T--1

,~.z2,w ~ O o v , ( - ~ d , n0 _> . . .~s-~ _> . . . _> ~ - ~ i = 0

recalling our definition of numerica l character we observe that

a - - 1 r - - 1 a - 1

r : . I2,w % (~ Op,(-fii) = (~ Op,(-fii)(9 (~ Op,(-2i) i : 0 i = 0 i = r

and so we can conclude as in the proof of theorem 1 that fq - fii+l _< 2 if i = 0 , . . . , s - 2 .

R e m a r k 4.2 It is not true, however, that we can prove fi{ - fii+l <-- 2 for the remaining values i = s - 1 , . . . , r - 2, because in these cases the proof of the theorem would lead to a curve of type (2e, e), e >_ s, containing 2 which is not a contradiction i f this curve is cut by a surface in p3 passing through the vertex.

We can now derive the b o u n d we are seeking for.

L e m m a 4.3 Let us consider 2s,~ = (mi)i=0 ....... 1 a character of length s and degree d. Let V~,.a the associated volume. Then,

V,~,,,~ < V,<~ I-

where V~7 ] . is the volume associated to the character

,tL? ~ = 1,~,}o_<,<,-,

t - 2 i + 2 s - 3 f o r O 2s(s + 1),its sectional irregularity q verifies

q < B(d, ~) ~~ v~,T,~

Proof: Let 2 ( 2 = X n C2) = {fii}i=0 ....... 1 be the numerical character of X. Then we want to prove tha t

On t h e s e c t i o n a l i r r e g u l a r i t y o f c o n g r u e n c e s 263

But a s t raightforward t rans la t ion of the analysis in lemmas 2.1, 2.2 and 2.3 together with the fact jus t proved: fii - fii+l _< 2, i = 0 , . . . , s - 2 finishes the proof of theorem 2. []

The bound jus t derived corresponds to Vm-~'~,d , associated to a character 2,m~ ~ =

{ f i i}o<i<~-l . Hence the irregulari ty may reach the bound for a.n. congruences whose character is this maximal one.

W h a t we poin ted out for the bicharacter of the a.n. congruences yields now a relat ionship between the bounds B ( d , d', s) and B(d, s). A n d we can conclude when the lat ter is sharp from tha t relation. If X is an a.n. congruence contained in a complex ~ of degree s, S = E N C3 and Y = X n C3 we have

s - 1

~ .z~ .~ = 0 o q ~ ( - ~ ' j , - ~ j ) 3 = 0

where Im'j - r n j l < 1, 0 <_ j < s - 1. Rest r ic t ing the projection to the preimages o f the two families of lines in Q2, we see tha t the {mj} and the {re'j} de termine the numerical characters x ( Z = X 93 c~) and x ( Z ' = X F1 c~'). If we restrict to a general hyperplane section of Q2, we recover the character 2 (Z = X FI C2). Now it is clear tha t

fii = m i Jr m~i, 0 < i < s - 1

Suppose tha t there is an a.n. congruence X , of postulat ion s such that

, t ( 2 = x n c~) = {~do_<,_<~-i

A priori we do not know the decomposi t ion

s--1

~.Z~,s = 0 0 q ~ ( - m ' j , -m~) j=0

But, if we restrict to congruences of maximal character , and after having a look at the calculat ion of q in terms of the c~ and a ' -characters , we see tha t if

with

L ~ ~ = {~,}0_<,_<,-,

t - 2 i + 2 s - 3 f o r 0 < i < r - 1 h i = t - 2 i + 2 s - 2 f o r r < i < s - 1

where d = st - r; 0 < r < s, then V~3~ coincides with the value of q for an a.n.

congruence of bidegree (d, d') as follows

1. e l = s t - r , 0 < r < s, r e v e r t :

t t d = s -~ - r, a n d d' = s 2

and the bicharacter of the congruence is

rr l a ~ t u t t i : t

• ( z ) = x~,~ = {~d0_<~<,-,; x (Z' ) = x,,~, = {~ d0_<~_<,-,

264 Giraldo, Sols

and s--1

3=0

w i t h m i = n i a n d m ' i = n ' i , O < i < s - 1 .

2. d = s t - r , O<_r < s, todd:

d - s ( t -1 ) andd'- s( t+l) 2 2

and the bicharacter of the congruence is

and

x ( z ) = x,~j ~ = {n,}o<,_<,_,; X(Z ' ) = X : T = {n',}o<_,<~-,

s - I

~.z~,~ : @ o q , ( - ~ ' ~ , -m~) j=O

and

being

�9 i f d = , s t - r , O < r < s w i t h t even

t - 2 i + 2 s - 4 fii = t - 2i + 2s - 3

t - 2i + 2s - 2

q <_ ~ .... (d, s)

for O < 2i < r - 2 f o r r - 2 < 2 i < _ r - 1 for r - 1 < 2 i < _ s - 1

where

with mi = ni and m'i = n'i, 0 2s(s + 1). Let X be an integral congruence of postulation >_ s, whose sectional irregularity reaches the bound B(d , s ) . Then X is a.n., it is contained in a complex of degree s and

d = - 1 , 0 , 1 (mod2s)

Furthermore, in all the possible cases there is a smooth congruence X like before whose sectional irregularity q verifies

q = B(d, s)

We can establish sharp bounds for arithmetically normal congruences

P r o p o s i t i o n 4.2 Let X be an a.n. congruence, of postulation >_ s and of degree > 2s(s + 1). Then their sectional irregularity is bounded by

�9 i f t i sodd, i f t ' = t + l a n d # = r + s we write

= 2st' - r', s ~_ r' < 2s

t ' - 2 i + 2 s - 4 f o r O < _ 2 i < r ' - 2 fi, = t' - 2i + 2s - 3 for r' - 2 < 2i <_ r' - I

t ' - 2 i + 2 s - 2 f o r r ' - l < 2 i < s - 1

Proof: It is enough to observe that the corresponding bicharacter is a maximal one and can be realized according to Proposition 3.2. []

Finally we can formulate the following

C o n j e c t u r e 4.1 The sectional irrcgularit 9 q of a congruence X of postulation > s and degree cl > 2s(s + 1) satisfies

q <_ B ... . (d, s)

R e m a r k 4.3 The relationship between the characters of Z and of Z and Z'

shows that if I d - d' t > s and d, d' > .s( s + 1)

~(d, s) < ~(d, d', s)

R e f e r e n c e s

[1] Arrondo, E - Sols, I.: Classification of smooth congruences of low degree, J. reine angew. Math. 393,199-219 (1989)

[2] Arrondo, E. - Sols, l.: On congruences of lines in the projective space, Mdm. Sac. Math. France 50, t .120 (1992)

[3] Fogarty, J.: Algebraic families on an algebraic surface, Amer.J.Math. 90,511-521 (1968)

[,1] Gross, M.: Surfaces of degree 10 in tile Grassmannian of lines in 3-space, J. reine angew. Math. 436 , 87-127 (1993)

[5] Gross, M.: The distribution of bidegrees of smooth surfaces in Gr(1, p3), Math. Ann. 292, 127-147 (1992)

[6] Gross, M.: Surfaces of bidegree (3, n) in Gr(t,W), Math. Zeit. 212 , 73-106 (i993)

[7] (/ruson, L. - Peskine, C.: Section plane d'une courbe gauche: postulation. Progress in Mathematics 24, Birkh/iuser, 1980

[8] Gruson, L. - Peskine, C.: Postulation des courbes gauches. Algebraic geometry - Open problems, (Ravello 1982), Leer. Notes in Math. 997, Berlin, Ileidelberg, New York 1982

266

[9]

[1o]

[11]

[12]

[13]

Giraldo, Sols

Gruson, L. - Peskine, C.: Genre des courbes de l'espace projectif. Pro- ceedings of Tromso conference on algebraic geometry, Lect. Notes in Math., 687, Berlin, Heidelberg, New York 1977, 31-59

Herngndez, R.- Sols, I.: On a family of rank 3 bundles on Gr(1, 3), J. reine angew. Math. 360, 124-135 (1985)

Herngndez, R. - Sols, I.: Line congruences of low degree, G6om6trie alg6brique et applications II: Singularitds et gdomftrie eomplexe, Paris 1987, 141-154

Laudal, O. A.: A generalized trisecant leinma. Proceedings of Tr0mso conference on algebraic geometry, Lect. Notes in Math. 687, Berlin- Heidelberg-New York 1977

Peskine, C.- Szpiro, L.: Liaison des vari4t6s alg~briques I, Invent. Math., 26, 271-302 (1974)

on the sectional irregularity of congruences

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